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An optical squeezed vacuum has zero mean field, nonzero mean photon number ${\sinh }^{2}r$, and unequal noise in two principal optical quadratures. With $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$, the vacuum variance is $1/4$ and $[X_1,X_2]=i/2$. In axes aligned with the squeezing,The ideal pure state saturates $\Delta X_1\Delta X_2=1/4$. A below-threshold degenerate optical parametric amplifier or oscillator can generate it: a coherent pump drives pair creation into a mode, and the pump phase selects the squeezed quadrature. Loss mixes in vacuum noise, while technical phase fluctuations mix the squeezed and antisqueezed axes.

For the stated vacuum–one-photon superposition, normalization requires $|ϵ|\le 1$. Direct Fock-basis matrix elements giveUsing $a{a}^{†}=a^{†}a+1$,Consequently there is no squeezing of the specified quadrature:It equals the vacuum uncertainty only at $ϵ=0$ and reaches $\sqrt{3}/2$ at $ϵ=\pm 1$. The curve is even in $ϵ$. The orientation matters: the same real superposition has $(\Delta X_1{)}^2=1/4-{ϵ}^2/2+{ϵ}^4$, which is below $1/4$ for $0<{ϵ}^2<1/2$. This does not turn it into a Gaussian squeezed vacuum; it simply illustrates quadrature squeezing of a vacuum–one-photon superposition.

For the printed beam splitter convention, expansion gives the operator identitiesThuswhere different input modes commute. The source's equality mixing an expectation on one side with an operator on the other is understood as this operator identity, or with expectation brackets on both sides. The mean difference reveals phase-sensitive mutual coherence between inputs; it is not by itself a photon-number correlation measurement of the usual Hanbury Brown–Twiss type. Repeated fluctuations, rather than one mean value, are needed to infer quadrature uncertainty.

For the independent coherent local oscillator, replace $b,b^{†}$ by $\beta ,{\beta }^{\ast }$ in the mean. The effective signal operator isTherefore $\varphi =0$ measures $X_2$, and $\varphi =-\pi /2$ measures $X_1$, each multiplied by $2|\beta |$. Choosing $\varphi =\pi /2$ measures $-X_1$. These phase choices follow the specified complex beam-splitter matrix; importing a different beam-splitter convention would shift them.

This is balanced homodyne detection. A stable strong local oscillator supplies the phase reference, the signal and oscillator meet at a 50:50 beam splitter, and matched photodetector currents are subtracted. Varying the reference phase selects different quadrature projections. Recording their distributions supplies more information than their means and supports optical homodyne tomography.

The strong-reference limit also needs a noise qualification. Define $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$, with $\vartheta =\varphi +\pi /2$. For independent signal and coherent reference, an exact ideal-detector calculation givesOne derivation writes $b=\beta +\delta b$, where $\delta b$ is in vacuum. The reference-amplitude term gives the first contribution; the remaining vacuum term contributes $⟨a^{†}a⟩$. Thus the normalized variance approaches the signal quadrature variance when the reference is sufficiently strong relative to the signal. A finite reference is not exactly a classical noiseless field.

A single detector can also carry quadrature information, but its mean alone still does not determine the variance. It measures $n_C=(n_a+n_b+I_{CD})/2$, retaining the large reference background and its fluctuations. For fixed signal moments and a large coherent reference,The additional vacuum-level term and the uncancelled reference power must be calibrated. In practice the reference may also have much larger classical intensity noise. Balanced subtraction cancels the common total-power term and strongly rejects common-mode technical noise while retaining the phase-sensitive difference. Detector imbalance, unequal efficiencies and reference phase noise limit that advantage. It is therefore the more direct and usually more sensitive quadrature-noise measurement, rather than a claim that a single-port measurement contains no useful information.